Connected sums of constant mean curvature surfaces in Euclidean 3 space

Abstract

Introduction and statement of the resultsAmongst the recent developments in the study of embedded complete minimal and constant mean curvature surfaces in R 3 is the realization that these objects are far more robust and flexible than is apparent from their Weierstrass representations. Our aim in this paper is to prove a 'gluing theorem', which states roughly that if two (appropriate) constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. We shall explain what we mean by appropriate, or at least give our preliminary interpretation of it, in the next paragraph. Throughout this paper, the acronym CMC shall mean a surface with constant mean curvature equal to one (or minus one depending on the orientation).The simplest context for our result is when we are given two orientable, immersed, compact CMC surfaces, Σ 1 and Σ 2 , with nonempty boundary. Suppose that we have applied a rigid motion to each of these surfaces so that 0 ∈ Σ 1 ∩ Σ 2 and T 0 Σ 1 = T 0 Σ 2 is the x y-plane. (These surfaces may intersect elsewhere, but that is irrelevant for our considerations.) We now define the orientation on these surfaces so that at 0 the oriented unit normal ν 1 of Σ 1 equals (0, 0, 1), while the oriented unit normal ν 2 of Σ 2 equals (0, 0, −1). Let us assume that with this orientation the two surfaces have the same mean curvature H 0 (so either H 0 = 1 or H 0 = −1 for both of the surfaces). We shall prove that there is a 'geometric connected sum' of these two surfaces, which may be thought of as a desingularization of this configuration. Moreover, the boundary of this desingularization will be the union of the boundaries of the Σ i , each possibly transformed by a small rigid motion.In order to state this first result rigorously, we make the following definition:Definition 1 A compact CMC surface Σ with boundary is said to be nondegenerate if there are no Jacobi fields on Σ which vanish on ∂Σ. Namely, if w : Σ −→ R is a C 2,α solution ofthen w = 0. Here A Σ is the second fundamental form of Σ.Theorem 1 (Connected sum theorem) Let Σ 1 and Σ 2 be two compact, smooth, immersed, orientable, nondegenerate CMC surfaces with boundary. Assume that these surfaces are positioned and oriented as above and have the same mean curvature H 0 . Then there exist an ε 0 > 0 and a one-parameter family of surfaces S ε , for ε ∈ (0, 4. The dilated surface ε −1 S ε converges in the C ∞ topology on any compact set to a catenoid with vertical axis.

Constant mean curvature surfaces with Delaunay ends

“…It is a consequence of the present work and our more recent paper [11] concerning connected sums of nondegenerate CMC surfaces that many more of these CMC moduli spaces contain nondegenerate elements and thus attain their correct dimension.…”

Generalized connected sum construction for scalar flat metrics

“…and the estimates (14), (15) and (16). Then, setting f (1) := −R (0) , we find another triple (λ (1) , u (1) , R (1) ) with the same properties as the first one and so on.…”

“…They have been used to understand solutions to problems arising from the geometry (minimal and constant mean curvature surfaces [13,14], constant scalar curvature metrics [9,12,15], and recently even Einstein metrics [1]) and from the physic (Einstein constraint equations [7] and [8]). However most of the results are concerned with the connected sum at points (point-wise connected sum), whereas the case of connected sum along a submanifold (generalized connected sum or fiber sum) has received less attention.…”

“…Notice that the estimate (17) and the estimate (26), with its analog forũ 2 , imply at once the estimate (14) in the statement of Lemma 1, namely there exists a positive constant A K ,n,γ ,α 1 ,α 2 such that…”

The dynamics theorem for CMC surfaces in $R^3$

“…The collection of properly embedded CM C surfaces with bounded second fundamental form is quite large and varied (see [4,10,11,15,16,17]). Many of these examples appear as doubly and singly-periodic surfaces.…”